Quantization in generalized coordinates III-Lagrangian formulation

Gary R. Gruber

F. J. Bloore, L. Routh

For pt.II, see abstr. A66629 of 1973. The authors show that the Frechet derivative, which they have previously used to formulate a quantum-mechanical version of Hamilton's principle of stationary action, suffers from certain defects. They set up a simpler, coordinate-free, formulation of quantum mechanics based on the Lie algebra of symmetric contravariant tensor fields on the configuration manifold of the system and re-express the principle of stationary action using this formalism. They find the same relation between Hamiltonian and Lagrangian as before, but a much larger class of allowable variations, namely those associated with and C infinity vector fields.

Gary R. Gruber

Hideki Kamo, Toshiharu Kawai

Schwinger’s variational principle is formulated for the multi-dimensional quantum system which corresponds to the classical system described by the Lagrangian Lc(\dotx,x)=(M/2)gij(x)\dotxi\dotxj-v(x). The c-number variations of coordinates and time are sufficient to give the laws of quantum mechanics. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relations are derived from this principle and there is no inconsistency. An appropriate choice of the Lagrangian operator is essential in our formulation. It is shown that an arbitrary point transformation is entitled to be called a canonical transformation. Ambiguities in the quantal Lagrangian are also discussed.

Nikolai Andreevich Tyurin

Ola A. Jarab’ah

Path integral formulation based on the canonical method is discussed. The Hamilton Jacobi function for regular Lagrangian is obtained using separation of variables method. This function is used to quantize regular systems using path integral method. The path integral is obtained as integration over the canonical phase space coordinates. One illustrative example is considered to demonstrate the application of our formalism.